You have 2 different systems of electricity. A 120/208 volt system is a 3 phase system. This means you have 3 wires wired together in a "Wye" or star fashion. The voltage reading between these wires is 208 volts. A 4th wire is attached to the center of the wye or star. This is where the 120 volts comes from. The formula is 208V/1.73= 120 volts.

A 120/240v system is derived from a transformer having a center tap on the 240 volt coil. This gives you your 1/2, or 120 volts.
Jim